What is Coordinates: Definition and 1000 Discussions

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

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  1. C

    Rindler Coordinates: Signals That Never Arrive

    Hi everybody, I know that there are a lot of threads in this forum about Rindler coordinates but none of them have helped me :confused: I'll explain you my problem. First of all, my coordinates (x^0,x) (Cartesian coord., where x^0=ct) are related to the Rindler coordinates (\omega ^0,\omega)...
  2. E

    Spherical Coordinates Question

    Homework Statement I'm feeling a bit ambivalent about my interpretation of spherical coordinates and I'd appreciate it if someone could clarify things for me! In particular, I'd like to know whether or not my derivation of the coordinates is legitimate. Homework Equations Considering...
  3. deedsy

    Divergence in cylindrical/spherical coordinates

    Homework Statement I'm just having trouble understanding a step in my notes from class.. We're talking about how to derive the divergence in other coordinate systems. Homework Equations So, we are deriving this divergence formula in spherical coordinates \oint \vec{A}\cdot d\vec{A} = \int...
  4. M

    Physical Interpretation of Coordinates in GR

    What is the relationship between the differentiable manifold that is space-time and the physical space around us? How does one relate the three seemingly Cartesian coordinates around us, those which we can measure out with a ruler, to the coordinates of the Lorentzian manifold? If i say, measure...
  5. P

    Units for Coordinates - Understanding the Debate

    I was reading "Time scales in the context of general relativity" Bernard Guinot, and a few other papers whose names I forget, and was surprised that there was apparently some desire by some physicists to give coordinates units. It seems that the current recommended practice is that...
  6. Ascendant78

    Mechanics in cartesian coordinates

    Homework Statement A cannon shoots a ball at an angle θ above the horizontal ground. (a) Neglecting air resistance, use Newton's second law to find the ball's position as a function of time. (Use axes with x measured horizontally and y vertically.) (b) Let r(t) denote the ball's distance...
  7. M

    Polar Coordinates [Finding the velocity]

    Homework Statement The projectile A is being tracked by the radar at O. At a given instant, the radar readings are θ = 30degrees, R = 2000m, dR/dt = 200 m/s, and d^2R/dt^2 = 20 m/s^2. Determine the speed of the projectile at that instant. THE ANSWER AT THE BACK IS 299.7m/s [PLEASE SEE...
  8. A

    Curvilinear basis in spherical polar coordinates

    Homework Statement As a part of my self study I am trying to find the covariant basis vectors in the spherical polar coordinates. Since I have never done anything like this before I would appreciate if someone could tell me whether I am on the rigth track. Homework Equations...
  9. squelch

    Torque about a point given coordinates in three dimensions

    Homework Statement Let \vec{F}=2\hat{i}-3\hat{j} act on an object at point (5,1,3). Find the torque about the point (4,1,0) Homework Equations \tau = \vec F \times \vec r The Attempt at a Solution Please tell me if my procedure is correct. Let the object occupy point A at (5,1,3) and let...
  10. E

    MHB Notation for vector coordinates in a given basis

    Sorry for a long post. I am looking for a clear and concise way to explain how to compute coordinates when changes of basis or linear operators are involved. I would like to avoid the summation notation as much as possible and use the definition of matrix multiplication only in the beginning...
  11. A

    Derivation of LLG equation in polar coordinates

    The torque contribution due to the uniaxial anisotropy is given by the equation below \frac{\Gamma}{l_m K} = (2 \sin\theta \cos\theta)[\sin\phi e_x - \cos\phi e_y] (3) This contribution can be taken in the LLG equation to derive the LLG equation in polar coordinates \frac{\partial...
  12. STEMucator

    Solving Homework: Polar Coordinates Issue on Volume

    Homework Statement My answer seems to differ from the books answer, so I'm wondering where something has gone wrong. Find the volume inside the prism bounded by the planes ##y = x##, ##y = 0##, ##x = \frac{a}{\sqrt{2}}##, ##z = 0## and the cone ##az = h\sqrt{x^2 + y^2}##. Homework...
  13. R

    Derivation of the Lagrangian for Rotating Polar Coordinates

    I'm reading Leonard Susskind's The Theoretical Minimum Vol. 1. 1. The problem: I'm on the section in which he asks the readers to derive the Lagrangian for a particle on a rotating carousel in polar coordinates. 2. Relevant ideas: The same Lagrangian in Cartesian coordinates is given as...
  14. M

    Confusion with Dot Product in Polar Coordinates with the Metric Tensor

    Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...
  15. C

    Geocenteric Equatorial Coordinates of the Sun

    I'm rather new here so please forgive me if this is answered somewhere else, but I was unable to find it while searching around. For some calculations I'm looking to perform I need to know the geocentric equatorial coordinates of the sun on a given date. However, the only place I know they...
  16. D

    Polar Coordinates, intersection of a cylinder with a spher

    Homework Statement Find the Volume of the solid that the cylinder ##r = acos\theta## cuts out of the sphere of radius a centered at the origin.Homework Equations The Attempt at a Solution I have defined the polar region as follows, $$D = \{ (r,\theta) | -\pi/2 ≤ \theta ≤ \pi/2 , 0 ≤ r...
  17. Dale

    Coordinate Charts vs Generalized Coordinates

    When you have a general coordinate chart on spacetime you have a lot of freedom to pick your coordinates, but you are always going to have 4 coordinates and each 4-tuple uniquely (in that chart) identifies one event in the manifold. When you are choosing generalized coordinates for a...
  18. schrodingerscat11

    Charge distribution of point charges in spherical coordinates

    Homework Statement Hi! This is not really a problem. I'm just confused on how to express the charge distribution of a set of point charges in spherical coordinates. From our discussion, ρ(\vec{r})=\sum\limits_{i=1}^N q_i δ(\vec{r}-\vec{r}') where \vec{r} is the position of the point where...
  19. M

    Why is the range of ø in spherical coordinates limited to 0 to π?

    Homework Statement In spherical coordinates (ρ,θ,ø); I understood the ranges of ρ, and θ. But ø, still eludes my understanding. Why is ø only from 0 to π, why not 0 to 2π??
  20. S

    MHB Convert to a equation in RECTANGULAR coordinates

    What am I doing wrong? Convert r = 2sin\theta to an equation in rectangular coordinates.. x^2 +y^2 = r^2 x^2 + y^2 = 2y x^2 + y^2 - 2y = 0 x^2 + y^2 - 2y - 1 = 1 x^2 + (y-1)^2 = 1 Coordinates are (0,1) yes?
  21. J

    Double dot product in Cylindrical Polar coordinates

    Hello, I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: 2W = σijεij Where σ and ε are symmetric rank 2 tensors. For cartesian coordinates it is really easy because the metric is just the identity matrix, hence: 2W = σxxεxx +...
  22. D

    2D quantum harmonic oscillator in cylindrical coordinates (radial part

    Dear kind helpers, actually I am not 100% sure whether this is the right place to post, as it is not a homework in the sense of an exercise sheet. But I think it could be because it feels pretty basic and that I should be able to solve it. Though I really searched for a solution but could not...
  23. WannabeNewton

    Series expansion tetrad Fermi coordinates

    Hi all. I'm working on a project that requires me to perform calculations in Fermi normal coordinates to certain orders, mostly 2nd order in the distance along the central worldline orthogonal space-like geodesics. In particular I need a rotating tetrad propagated along the central worldline...
  24. C

    Rotation of coordinates (context of solving simple PDE)

    If you rotate your rectangular coordinate system (x,y) so that the rotated x'-axis is parallel to a vector (a,b), in terms of the (x,y) why is it given by x'=ax+by y'=bx-ay I got x'=ay-bx, y'=by+ax from y=(b/a)x. By the way this is from solving the PDE aux+buy=0 by making one of the...
  25. S

    Creating 3d coordinates from stereoscopic images

    Hi, I'm working on a project that would take a 3d image using stereocopic camera and would record the depth and the 2d (x1,y1) , (x2,y2) coordinates of a single point in the image. The depth is found using the focal point, disparity, and the distance between the difference the camera sees on...
  26. Mr-R

    Spherical coordinates metric

    Dear all, As I was reading my book. It said that the line element of a particular coordinate system (spherical) in R^{3} is so and so. Then it said that the metric is flat. I don't get how the metric is flat in spherical coordinate. Could someone shed some light on this please? Thanks
  27. S

    Cristoffel Symbol of spherical coordinates

    I just derived the 3-D Cristoffel symbol of the 2nd kind for spherical coordinates. I don't think I made any careless mistakes, but once again, I just want to verify that I am correct and I can't find any place on line that will give me the components of the symbol so I can check myself. Here...
  28. S

    Metric Tensor in Spherical Coordinates

    I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...
  29. G

    Discretization in cylindrical coordinates, unit thickness for azimuth?

    I am setting up a numerical simulation from a 2D discretization of the heat equation in cylindrical coordinates. my spatial variables are radius (r), height (z), and azimuth (ø). The assumption is that there is no gradient along the azimuth direction (if temperature is T then dT/dø = 0)...
  30. C

    Generating the function from given coordinates and slopes

    hi fellas, i have been given a graph from which i can extract the coordinates and the slopes but all i need is to generate the function this graph represents. can you suggest me any manual procedure to do this mathematically or do i have to use software to generate polynomial functions? if...
  31. C

    Cartesian or Polar Coordinates to store intergalactic objects in DB?

    So I'm wondering, should I use Cartesian or Polar Coordinates to store intergalactic objects in DB? I'm currently prototyping a game idea that can be oversimplified as a spaceship simulator in infinite space. I'm considering grouping objects together so that they have a "parent super-space"...
  32. R

    Using polar coordinates to determine the limit

    Lim (x, y)->(0,0)(X^3+y^3)/(x^2+y^2) The answer is -1, but I can't get it there. Here is what I did. ((Rcosx)^3 +(rsinx)^3)/((rcosx)^2+(rsinx)^2) Then by factoring out a r squared from top and bottom I'm left with a denominator of (sin^2(x ) + cos^2 (x)) which simplifies to 1. And a numerator...
  33. J

    Equation general of conic in polar coordinates

    The conic equation has 2 versions in cartesian coordinates: The general: ##Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0## And the parametric: ##y^2 = 2px + (e^2-1)x^2## In polar coordinates, I known just the parametric: ##r = \frac{p}{1+e\cos(\theta)}## But exist a general form too?
  34. adjacent

    Finding Coordinates of A and B in an Equilateral Triangle

    Homework Statement The points O,A and B are vertices of an equilateral triangle. Find a and b O=(0,0) A=(a,11) B=(b,37) Homework Equations ##c^2=a^2+b^2## The Attempt at a Solution Let AB =c Then ##c=\sqrt{(a-b)^2+(11-37)^2}=\sqrt{(a-b)^2+676}## Since it is an equilateral triangle...
  35. Saitama

    MHB Solve for the coordinates of square

    Three unit circles $C_1$, $C_2$ and $C_3$ in a plane have the property that each circle passes through the centres of the other two. A square $ABCD$ surrounds the three circles in such a way that each of its four sides is tangent to at least one of $C_1$,$C_2$ and $C_3$. $A=(0,0)$, $B=(a,0)$...
  36. R

    Is it possible to convert 2D coordinates of point to 3D form ?

    Hello everyone ,i have captured car positons at differents frames.http://www.imagesup.net/pt-7140205392313.png%5D%5BIMG%5Dhttp://www.imagesup.net/dt-7140205392313.png Suppose car's(left side car which is coming towards us) centroid is at video frame1 is P(x1,y1) and Q(x2,y2) at video frame4...
  37. evinda

    MHB Partial derivatives-polar coordinates

    Hello! :) From the relations: $$\partial_{r}=\cos \theta \cdot \partial_{x}+ \sin \theta \cdot \partial_y$$ $$\partial_{\theta}=-r \sin \theta \cdot \partial_x+ r \cos \theta \cdot \partial_y$$ we get: $$\partial_y=\sin \theta \cdot \partial{r}+\frac{\cos \theta}{r} \cdot \partial_{...
  38. S

    Poisson's Equation in Cylindrical Coordinates

    Homework Statement Homework Equations Possions Equation and boundary conditions... The Attempt at a Solution First Part that I think is right... However when I try and apply the boundary conditions ie V(a)=V(r)=0... I can't get the answer! And for the last...
  39. M

    Need free program to plot expressions in polar coordinates

    Homework Statement Need a free program to plot expressions in polar coordinates. For example, I want to plot the equipotentials for an expression in polar coordinates of the potential for a dipole charge, 4q and -q separated by a distance L. Homework Equations V=kq(4/r1 - 1/r), where r12...
  40. N

    Set of vectors whose coordinates are integer (is a subspace?)

    Homework Statement For a set of vectors in R3, is the set of vectors all of whose coordinates are integers a subspace?The Attempt at a Solution I do not exactly understand if I should be looking for a violation or a universal proof. If x,y, z \in Z then x,y,z can be writted as...
  41. M

    How Do You Calculate θ_dot in Polar Coordinates?

    Homework Statement A particle moves with const speed v along the curve r(θ) = a(1+cos θ). Starting with the general expression for the velocity vector v in polar coordinates solve for θ_dot in terms of v, k, and θ. What does the sign of θ_dot signify?Homework Equations v = r_dot*r_hat +...
  42. O

    MHB Solving Polar Coordinates in System of Equations

    given that x'=f(x,y) y'=g(x,y) iff the vector function (r, θ) is a sloution of the system r'=f(rcosθ,rsinθ)cosθ +g(rcosθ,rsinθ)sinθ am trying to show that this is true but i just don't get where the sinθ and cosθ come from, how do i get to that
  43. B

    MHB Triple Integrals in Spherical Coordinates

    Hi all, I'm not sure how to get the boundaries in terms of both the spherical and cylindrical coordinates for this question. Here are the boundaries we were given in the solution. How was \frac{\pi}{4} for φ and \frac{1}{\sqrt{2}} for r obtained? Thanks!
  44. PsychonautQQ

    Question about cylindrical Coordinates

    I'm confused why when using cylindrical coordinates three unit vectors are needed. My book says that the three unit vectors are one for the radial direction which is bound to the xy plane and then a unit vector in the z direction. It goes on to say that there is another unit vector associated...
  45. J

    MHB How do I Find the Vector Coordinates to Solve for Trihedral Angle?

    Hi everyone, Here's the problem I have. Given two unit vectors A, B and angle φ between them. Find the coordinates (in 3D) of a unit vector C so that the angles between C and A,B be α and β respectively. α + β => φ and α + β + φ <= 360° It looks trivial to me and yet here I am asking for...
  46. N

    Taking partial derivative in polar coordinates

    In a problem that requires converting from cartesian to polar coordinates, I need to take \frac{dr}{dx}. I tried doing it two different ways but getting two completely different answers.. Method 1: r=\sqrt{x^2+y^2} \frac{dr}{dx}=\frac{1}{2}\frac{1}{\sqrt{x^2+y^2}}2x \;\; =...
  47. T

    Double Integrals using Polar Coordinates

    Homework Statement ∫∫Rarctan(y/x) dA, where R={(x,y) | 1\leqx2+y2\leq4, 0\leqy\leqx Homework Equations x=rcos(θ) y=rsin(θ) x2+y2=r2 The Attempt at a Solution I know that the range of r is 1 to 2 but I can't figure out how to change the second part into θ. If I change y and x to...
  48. H

    Finding Potential (Spherical coordinates )

    1. . An electric dipole located at the origin in free space has a moment p = 3ax −2ay +az nC·m. Find V at r = 2.5, θ =30◦, φ =40◦. I find it difficult to solve when its in spherical co-ordinates.2.Relevent Eq V =P.(r-r')/( 4∏ε|r−r'|2)(|r-r'|)I am confused how to find a unit vector on spherical...
  49. M

    Divergence of curl in spherical coordinates

    Hey pf! I was thinking about how div(curl(f)) = 0 for any vector field f. However, is this true for div and curl in spherical coordinates? It doesn't seem to be. If not, what needs to happen for this to be true in spherical coordinates?? Thanks all!
  50. J

    Write a triple integral in spherical coordinates

    Homework Statement Write a triple integral in spherical coordinates that represents the volume of the part of the sphere X^2+Y^2+Z^2=16 that lies in the first octant(where x,y, and z are coordinates are all greater than or equal to zero) Homework Equations So i know this is in...
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